CAN SOMEONE PLEASE SHOW ME EACH STEP I HAVE HAD THIS ONE ANS

CAN SOMEONE PLEASE SHOW ME EACH STEP. I HAVE HAD THIS ONE ANSWERED BEFORE BUT I NEED TO UNDERSTAND EACH STEP OF THE PROBLEM.

Solution

it is true for n=1 (obvious!)

let it is true for n = k

i.e. , 1 + 3 + 3^2 + .....+ 3^k = (3^k+1 -1)/2

add 3^(k+1) on both sides

i,e. , 1+ 3+ .... + 3^k + 3^(k+1) = 3^k+1 (1+0.5) - 0.5

= (3^k+2 - 1) / 2

hence it is true for n = k+1

therefore by mathematical induction , it is true for all n>0

put n =15

answer is (3¹⁶ - 1)/2